Table Of ContentAdvanced Mathematical Economics
Astheintersectionbetweeneconomicsandmathematicscontinuestogrowinboth
theoryandpractice,asolidgroundinginmathematicalconceptsisessentialforall
seriousstudentsofeconomictheory.
Inthisclearandentertainingvolume,RakeshV.Vohrasetsoutthebasicconcepts
of mathematics as they relate to economics. The book divides the mathematical
problems that arise in economic theory into three types: feasibility problems,
optimality problems and fixed-point problems. Of particular salience to modern
economic thought are sections on lattices, supermodularity, matroids and their
applications.Inadeparturefromtheprevailingfashion,muchgreaterattentionis
devotedtolinearprogramminganditsapplications.
Ofinteresttoadvancedstudentsofeconomicsaswellasthoseseekingagreater
understandingoftheinfluenceofmathematicson‘thedismalscience’.Advanced
MathematicalEconomicsfollowsalongandcelebratedtraditionoftheapplication
ofmathematicalconceptstothesocialandphysicalsciences.
Rakesh V. Vohra is the John L. and Helen Kellogg Professor of Managerial
Economics and Decision Sciences at the Kellogg School of Management at
NorthwesternUniversity,Illinois.
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Routledgeadvancedtextsineconomicsandfinance
FinancialEconometrics
PeijieWang
MacroeconomicsforDevelopingCountries2ndedition
RaghbendraJha
AdvancedMathematicalEconomics
RakeshV.Vohra
AdvancedEconometricTheory
JohnS.Chipman
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Advanced Mathematical
Economics
Rakesh V. Vohra
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Firstpublished2005
byRoutledge
2ParkSquare,MiltonPark,Abingdon,OxonOX144RN
SimultaneouslypublishedintheUSAandCanada
byRoutledge
270MadisonAve,NewYork,NY10016
RoutledgeisanimprintoftheTaylor&FrancisGroup
This edition published in the Taylor & Francis e-Library, 2009.
To purchase your own copy of this or any of Taylor & Francis or Routledge’s
collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.
©2005RakeshV.Vohra
Allrightsreserved.Nopartofthisbookmaybereprintedor
reproducedorutilizedinanyformorbyanyelectronic,
mechanical,orothermeans,nowknownorhereafter
invented,includingphotocopyingandrecording,orinany
informationstorageorretrievalsystem,withoutpermissionin
writingfromthepublishers.
BritishLibraryCataloguinginPublicationData
Acataloguerecordforthisbookisavailable
fromtheBritishLibrary
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ISBN 0-203-79995-X Master e-book ISBN
ISBN 0-203-68209-2 (Adobe ebook Reader Format)
ISBN0–415–70007–8(hbk)
ISBN0–415–70008–6(pbk)
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Contents
Preface viii
1 Thingstoknow 1
1.1 Sets 1
1.2 Thespaceweworkin 1
1.3 Factsfromrealanalysis 2
1.4 Factsfromlinearalgebra 6
1.5 Factsfromgraphtheory 9
2 Feasibility 13
2.1 Fundamentaltheoremoflinearalgebra 13
2.2 Linearinequalities 15
2.3 Non-negativesolutions 15
2.4 Thegeneralcase 19
2.5 Application:arbitrage 20
2.6 Application:co-operativegames 24
2.7 Application:auctions 25
3 Convexsets 33
3.1 Separatinghyperplanetheorem 34
3.2 Polyhedronsandpolytopes 40
3.3 Dimensionofaset 46
3.4 Propertiesofconvexsets 47
3.5 Application:linearproductionmodel 49
4 Linearprogramming 53
4.1 Basicsolutions 56
4.2 Duality 60
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vi Contents
4.3 Writingdownthedual 64
4.4 Interpretingthedual 64
4.5 Marginalvaluetheorem 68
4.6 Application:zero-sumgames 69
4.7 Application:Afriat’stheorem 72
4.8 Integerprogramming 75
4.9 Application:efficientassignment 78
4.10 Application:Arrow’stheorem 80
5 Non-linearprogramming 87
5.1 Necessaryconditionsforlocaloptimality 89
5.2 Sufficientconditionsforoptimality 95
5.3 Envelopetheorem 100
5.4 Anasideonutilityfunctions 103
5.5 Application:marketgames 105
5.6 Application:principal–agentproblem 109
6 Fixedpoints 117
6.1 Banachfixedpointtheorem 117
6.2 Brouwerfixedpointtheorem 118
6.3 Application:Nashequilibrium 125
6.4 Application:equilibriuminexchangeeconomies 127
6.5 Application:Hex 133
6.6 Kakutani’s fixedpointtheorem 136
7 Latticesandsupermodularity 142
7.1 Abstractlattices 149
7.2 Application:supermodulargames 151
7.3 Application:transportationproblem 152
7.4 Application:efficientassignmentandthecore 154
7.5 Application:stablematchings 159
8 Matroids 163
8.1 Introduction 163
8.2 Matroidoptimization 165
8.3 Rankfunctions 166
8.4 Deletionandcontraction 169
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Contents vii
8.5 Matroidintersectionandpartitioning 170
8.6 Polymatroids 175
8.7 Application:efficientallocationwithindivisibilities 179
8.8 Application:Shannonswitchinggame 186
Index 191
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Preface
Iwantedtotitlethisbook‘LeisureoftheTheoryClass’.Thepublishersdemurred.
Mysecondchoicewas‘Feasibility,OptimalityandFixedPoints’.Whileaccurate,
itdidnotidentify,asthepublishernoted,theintendedaudience.Wesettledatlast
ontheanodynetitlethatnowgracesthisbook.Asitsuggests, thebookisabout
mathematics.Thequalifier‘advanced’signifiesthatthereadershouldhavesome
mathematical sophistication. This means linear algebra and basic real analysis.1
Chapter 1 provides a list of cheerful facts from these subjects that the reader is
expectedtoknow.Thelastwordinthetitleindicatesthatitisdirectedtostudents
ofthedismalscience.2
Threekindsofmathematicalquestionsarediscussed. Givenafunctionf and
asetS,
• Findanx suchthatf(x)isinS.Thisisthefeasibilityquestion.
• Findanx inS thatoptimizesf(x).Thisistheproblemof optimality.
• Findanx inS suchthatf(x)=x;thisisthefixedpointproblem.
These questions arise frequently in Economic Theory, and the applications
describedinthebookillustratethis.
Thetopicscoveredarestandard.Exceptionsarematroidsandlattices.Unusual
forabooksuchasthisistheattentionpaidtoLinearProgramming.Itiscommon
amongstcyclopeanEconomiststodismissthisasaspecialcaseofKuhn–Tucker.A
mistakeinmyview.Ihopetopersuadethereader,byexample,ofthesame.Another
unusualfeature,Ithink,aretheapplications.Theyarenotmerelycomputational,
i.e.,thisishowoneusesTheoremXtocomputesuchandsuch.Theyaresubstantive
piecesofeconomictheory. Thehomeworkproblems, asmystudentswillattest,
arenotforthefainthearted.
Ofthemakingofbooksthereisnoend.3Theremarkisparticularlytrueforbooks
devotedtotopicsthatthepresentonecovers.So,someexplanationisrequiredof
howthisbookdiffersfromothersofitsilk.
The voluminous Simon and Blume (1994), ends (with exceptions) where this
bookbegins.InfactaknowledgeofSimonandBlumeisagoodprerequisitefor
thisone.
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Preface ix
Thethick,square,Mas-Colleletal.(1995)containsanappendixthatcoversa
subsetofwhatiscoveredhere.Thetreatmentisnecessarilybrief,omittingmany
interestingdetailsandconnectionsbetweentopics.
Sundaram’sexcellent‘FirstCourseinOptimization’(1996)isperhapsclosestof
themorerecentbooks.Buttherearecleardifferences.Sundaramcoversdynamic
optimizationwhilethisbookdoesnot.Ontheotherhand,thisbookdiscussesfixed
pointsandmatroids,whileSundaramdoesnot.
This book is closest in spirit to books of an earlier time when, giants, I am
reliably informed, walked the Earth. Two in particular have inspired me. The
first is Joel Franklin’s ‘Methods of Mathematical Economics’ (1980), a title
that pays homage to Courant and Hilbert’s celebrated ‘Mathematical Methods
of Physics’(1924). Franklin is a little short on the economic applications of the
mathematicsdescribed.However,theinformalanddirectstyleconveyhisdelight
inthesubject.ItisadelightIshareandIhopethisbookwillinfectthereaderwith
thesame.
The second is Nicholas Rau’s ‘Matrices and Mathematical Programming:An
IntroductionforEconomists’.Rauisformal,preciseandstartlinglyclear.Ihave
triedtomatchthisstandard,goingsofarastofollow,insomecases,hisexposition
ofproofs.
ThebookbeforeyouistheoutgrowthofaPhDclassthatallgraduatestudents
inmydepartmentmusttakeintheirfirstyear.4 Itsrootshowevergobacktomy
saladdays.Ihavelearntanenormousamountfromteachersandcolleaguesmuch
ofwhichinfusesthisbook.Innoparticularorder,theyareAilsaLand,SaulGass,
Bruce Golden, Jack Edmonds, H. PeytonYoung, Dean Foster, Teo Chung Piaw
andJamesSchummer.
Four cohorts of graduate students at Kellogg and other parts of Northwestern
have patiently endured early versions of this book. Their questions, both
puzzled and pointed have been a constant goad. I hope this book will do them
justice.
Denizens of MEDS, my department, have patiently explained to me the finer
points of auctions, general equilibrium, mechanism design and integrability. In
returnIhavesubjectedthemtoendlessspeechesabouttheutilityofLinearPro-
gramming.Thebookisasmuchareflectionofmyownhobbyhorsesasthespirit
ofthedepartment.
This book could not have been written without help from both my parents
(FaqirandSudeshVohra)andin-laws(HarishandKrishnaMahajan).Theytook
it in turns to see the children off to school, made sure that the fridge was full,
dinner on the table and the laundry done. If it continues I’m in clover! My
wife, Sangeeta took care of many things I should have taken care of, including
myself.
Piotr Kuszewski, a graduate student in Economics, played an important role
in preparing the figures, formatting and producing a text that was a pleasure to
lookat.
Finally, this book is dedicated to my childrenAkhil and Sonya, who I hope
will find the same joy in Mathematics as I have. Perhaps, like the pilgrims in
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